A physical analysis of sap flow dynamics in trees

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Martti Perämäki

Department of Forest Ecology, University of Helsinki, Finland

Academic dissertation

To be presented, with the permission of

the Faculty of Agriculture and Forestry of the University of Helsinki, for public criticism

in Lecture Hall 2 of the Viikki Info Centre Korona, Viikinkaari 11, Helsinki, on 17^th June 2005, at 12 noon.

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A physical analysis of sap flow dynamics in trees Martti Perämäki

Dissertationes Forestales 2

Työn ohjaajat/ Supervisors:

Prof. Pertti Hari, Department of Forest Ecology, University of Helsinki Doc. Eero Nikinmaa, Department of Forest Ecology, University of Helsinki Prof. Timo Vesala, Division of Atmospheric Physics, Department of Physical

Sciences, University of Helsinki

Esitarkastajat/ Reviewers:

Prof. N. Michele Holbrook, Department of Organismic and Evolutionary Biology, Harvard University, USA

Doc. Hanna Vehkamäki, Division of Atmospheric Physics, Department of Physical Sciences, University of Helsinki

Vastaväittäjä/ Opponent:

Dr. Risto Sievänen, Finnish Forest Research Institute

ISSN 1795-7389 ISBN 951-651-101-5

2005 Publishers:

The Finnish Society of Forest Science Finnish Forest Research Institute

Faculty of Agriculture and Forestry of the University of Helsinki Faculty of Forestry of the University of Joensuu

Editorial Office:

The Finnish Society of Forest Science Unioninkatu 40A, 00170 Helsinki, Finland http://www.metla.fi/dissertationes

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ABSTRACT

The aim of this thesis was to analyze the water dynamics of trees by using a dynamic modeling approach. The work is of cross-disciplinary nature: tree water issues which is a subject of whole plant physiology is examined by means of physics. From physical principles five different models were derived. First, to study the sap flow and the water pressure dynamics in the xylem of tree stem, the effect of embolism on the sap flow, and the recovery of embolized conduits; and second, to analyze a sap flow measuring system based on the heat balance method.

Sap flow and water pressure dynamics are analyzed with two models that are based on the relation between xylem water tension and changes in the diameter of sapwood. This new approach is advantageous because it offers a way to compare the model predictions with the easily measured diameter changes in intact trees. The model results give new in- sight into the water dynamics in the stem, stating that pressure propagation is fast, but time lags of a few minutes do exist. These time lags are related to the dimensions and radial elasticity of the stem, and to the sapwood permeability. The results are in agreement with the cohesion-tension theory but partly contradict the pipe model theory of a plant form.

The embolism recovery model presents a quantitative analysis of the processes that have been suggested to recover embolized conduits while the water in surrounding conduits is under negative pressure. The model analysis reveals that under normal physiological cir- cumstances, e.g., normal xylem water tension, osmotic pressure of living cells, or diffusion distances, the refilling process is possible if the two sides of the same living cell have different transport properties for solutes.

The model for analyzing the performance of a sap flow measuring system reveals that the method, that assumes homogenous temperature field inside the stem, is not appropriate for stems with a diameter larger than a few centimeters.

This study has shown that relatively simple models, based on physical principles, are useful in increasing our understanding of the processes and behavior of natural objects.

Furthermore, model analysis leading to better understanding of dynamic systems, can guide forthcoming research and enhance the development of new instrumentation.

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ACKNOWLEDGEMENTS

It was one of those long evenings in Hyytiälä in summer 1995 when I, as a young forestry student, ran into Pepe Hari in the surroundings of the greenhouse. He was quite busy with the coming great opening ceremony of the SMEAR II station but I managed to ask if he had some subject for a master’s thesis. He replied that there are three topics of which one is the most interesting. ‘What’s that one?’ I asked. ‘Pitäisi selvittää, miten vesi lillii puussa’, (We should find out how water is dwelling or soaking inside a tree) he replied.

Now, after ten years, we are looking forward to the coming 10^th Anniversary of the SMEAR II station. During those years I have had the privilege to engage with the efforts of the group working with tree water issues. The members of this superb group are Pepe and Eero Nikinmaa from Department of Forest Ecology and Timo Vesala, Sanna Sevanto and Teemu Hölttä from Division of Atmospheric Physics, Department of Physical Sciences.

I thank the whole group and especially my supervisors Pepe, Eero and Timo, who have given support and valuable comments during the whole process. In addition, they have taught me a lot of scientific realism, forest ecology, environmental physics and the song of Lesser Whitethroat (Sylvia curruca). With them I have had the most interesting discussions about birds and the some other essential topics of life.

I express my gratitude to Erkki Siivola and Toivo Pohja, whose technical constructions have been invaluable in the field measurements, and to Hannu Ilvesniemi as a co-author of the first paper. Tuuli Timonen and Pirkko Harju (Botanical Museum, Univ. of Helsinki) are acknowledged for the microscope photos of Scots pine cells.

Missy Holbrook and Hanna Vehkamäki who as reviewers have carefully read this thesis are gratefully acknowledged. I appreciate their evaluations and comments.

Special thanks to my colleagues at our department for sharing this burden: Jukka Pum- panen, Eija Juurola, Pasi Kolari, Maarit Raivonen, Nina Tanskanen, Petteri Vanninen and Roope Kivekäs among others. Muchas gracias a Nuria Altimir y Albert Porcar. Albert is also acknowledged for the language revision of the summary part of this thesis. I appreciate the inspiring atmosphere among the people in both the Department of Forest Ecology, headed by Pasi Puttonen, and the Division of Atmospheric Sciences, headed by Markku Kulmala.

Financial support from the Academy of Finland and Tekes Technology Development Centre, Finland is acknowledged.

Warm thanks to my relatives for all kinds of support.

My folks, Laura, Jussi, Elli and Eero, have been optimistic during all these years no matter what the speed of this process occasionally has been. Thanks for your patience.

May, 2005 Martti Perämäki

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LIST OF ORIGINAL ARTICLES

This thesis consists of an introductory review followed by five research articles. In the review the articles are referred to using Roman numbers. They are reproduced with the permission of the publisher of the journals.

I. Martti Perämäki, Eero Nikinmaa, Sanna Sevanto, Hannu Ilvesniemi, Erkki Siivola, Pertti Hari and Timo Vesala. 2001. Tree stem diameter variations and transpiration in Scots pine: an analysis using a dynamic sap flow model. Tree Physiology 21: 889-897.

II. T. Hölttä, T. Vesala, M. Perämäki and E. Nikinmaa. 2002. Relationships between Embolism, Stem Water Tension, and Diameter Changes. Journal of Theoretical Biology 215: 23–38.

III. Timo Vesala, Teemu Hölttä, Martti Perämäki and Eero Nikinmaa. 2003. Refill- ing of a hydraulically isolated embolized vessel: Model calculations. Annals of Botany 91: 419-428.

IV. Martti Perämäki, Timo Vesala and Eero Nikinmaa. 2001. Analysing the applica- bility of heat balance method for estimating sap flow in boreal forest conditions.

Boreal Environment Research 6: 29-46.

V. Martti Perämäki, Timo Vesala and Eero Nikinmaa. 2005. Modeling the dynam- ics of pressure propagation and diameter variation in tree sapwood. Tree Physiol- ogy 25: nnn-nnn (in press).

Martti Perämäki developed and implemented the models and produced all the simulation results in studies I, IV and V. He was also responsible for the field measurements, data analysis and literature searches in these studies. In study II the sap flow sub-model was based on the model in Study I. The embolism sub-model was developed by Teemu Hölttä and Martti Perämäki. In addition, Martti Perämäki participated in the discussions and com- mented on the manuscripts in studies II and III. He is the main author of articles I, IV and V.

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LIST OF TERMS

Air-seeding Mechanism leading to embolism where a gas bubble is pulled into the cell lumen through a pit in the cell wall.

Aquaporin Cell membrane structure controlling water and solute flow through cell membrane

Bowen ratio Ratio of sensible to latent heat.

Cambium Thin layer of meristematic tissue between the phloem and the xylem of a stem. Produces phloem and xylem.

Cavitation Process where gas bubbles are formed in liquid phase in regions under low pressure.

Cell lumen Cavity, which the cell walls enclose.

Conduit Water-conducting cell in trees;

vessels in broad-leaved trees, tracheids in conifers.

Embolism Occlusion of a conduit by a gas bubble.

Evaporation Conversion of water to water vapor from the soil.

Evapotranspiration The combined evaporation from the soil surface and

transpiration from plants. Represents the transport of water from the earth back to the atmosphere.

Interconduit pit Structure in the cell wall that permits water flow between conduits but prevents gas bubbles to move from one conduit to another.

Latent heat The energy stored in water vapor in transpiration and evaporation.

Meristem Embryonic tissue, undifferentiated, actively dividing and growing cells.

Middle lamella First layer formed during cell division. The outer wall of the cell. Shared by adjacent cells.

Parenchyma cell General-purpose cell often capable of photosynthesis, and some conduction. Frequently a storage type cell for water and reserve foods.

Permeability Degree to which a solid allows the passage of a fluid through it.

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Phloem Part of a vascular bundle consisting of sieve tubes,

companion cells, parenchyma and fibers. Transports assimilates.

Reflection coefficient Measure of the relative permeability of a membrane to a particular solute.

Sap Water flowing in xylem conduits.

Sensible heat Heat absorbed or transmitted by a substance during a change of temperature, which is not accompanied by a change of state.

Tracheid Elongated xylem cell having lignified cell walls.

Transports water and offers mechanical support for the stem.

Transpiration Conversion of water to water vapor, through the plants.

Turgor Large positive internal pressure of living plant cells giving rise to mechanical rigidity of the cells.

Vessel Water conducting system in the xylem consisting of a column of cells.

Water tension Negative hydrostatic pressure.

Xylem Collective name for the cells, vessels, and fibres forming.

the harder portion of the fibrovascular tissue; the wood.

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LIST OF SYMBOLS AND CONSTANTS

Symbol Unit Description

A m² stem cross-section area

E_l mg m^-2 s^-1 leaf-specific transpiration

E_r Pa radial elastic modulus (elasticity) of sapwood

ET mm d^-1 evapotranspiration rate

G W m^-2 heat flux density into the soil substrate

GEP g C a^-1 gross ecosystem production, the total photosynthetic production of plants

H W m^-2 sensible heat flux density

k m² permeability of sapwood

A_l m² leaf area

LE W m^-2 latent heat flux density

p Pa water pressure

P_in W heating power of sap flow sensor based on SHB method

PAR µmol m^-2 s^-1 photosynthetically active electromagnetic radiation wavelength 400-700 nm

Q kg s^-1 sap flow rate

Q_r W radial heat flux in SHB method

Q_v W vertical heat flux in SHB method

θ kg m^-3 sapwood water content

r m stem radius

r_hw m radius of heartwood in stem

R_n W m^-2 net radiation

RH % relative humidity

T °C temperature

VPD Pa vapor pressure deficit of water vapor in air c 4.18 kJ kg^-1 heat capacity of liquid water

D_w 2.57×10^-5 m² s^-1 diffusion coefficient of water vapor in air D_c 1.60×10^-5 m² s^-1 diffusion coefficient of carbon dioxide in air g 9.81 kg m s^-2 acceleration due to gravity

γ 0.073 N m^-1 surface tension of water η 0.001 Pa s (20°C)dynamic viscosity of water ρ 1000 kg m^-3 density of liquid water

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1. INTRODUCTION

1.1 Distribution of forests in the world

Water stress is the most common limitation to growth of vegetation (Kozlowski et al., 1991). Especially, in the case of trees, the availability of water has a great influence on their performance. Figure 1 shows monthly gross ecosystem productivity (GEP) vs. evapotranspiration (ET) of several evergreen coniferous (a) and deciduous broadleaf (b) forest stands all over the world (Law et al. 2002). The figures indicate clear positive regression of productivity of plants on evapotranspiration.

Figure 1. Monthly gross ecosystem production (GEP) as a function of evapotranspiration (ET) in (a) evergreen coniferous forest and (b) deciduous broadleaf forest, Law et al. 2002. Reprinted with the permission of Agricultural and Forest Meteorol- ogy.

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From the figures above it could be concluded that forests could be most successful in areas without water deficit. This is actually the case. The maps in Figures 2 and 3 show that forest is the natural vegetation type all over the world where potential evaporation and transpiration do not exceed precipitation and where annual average temperature is adequate.

Figure 2 presents the global distribution of forests and Figure 3 illustrates aridity zones of the Earth. Arid denotes a region or climate characterized by very low rainfall, often sup- porting only desert vegetation. Aridity is the degree to which climate lacks effective, life- promoting moisture; opposite of humidity, in climate sense. The humid zone is most exten- sive, covering about 46.5 million km² (WRI 2002). As the figures show, almost all the humid areas are covered with forest (38.7 million km² in 2000, FAO 2001) and, vice versa, all the areas, which are covered with forest, are humid.

Figure 2. The global forests distribution. Dark-green: closed forest, more than 40 per cent covered with trees more than 5 metres high. Mid-green: open (10-40 per cent coverage) and fragmented forest. Light-green: other woodland, shrubland and bushland (IUCN 2005, source FAO 2001).

Figure 3. The aridity zones of the world (WRI 2002).

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Maximum tree height has been reported to depend on water availability (Ryan and Yo- der 1997, Koch et al. 2004). The biggest trees in the world are all living in extremely humid areas: Coastal redwood (Sequoia sempervirens (D.Don) Endl.), which can exceed height of 100 m and Douglas-fir (Pseudotsuga menziesii (Mirb.) Franco) (100m) in Pacific North- west, USA, where annual precipitation may exceed 3000 mm (Burns and Honkala 1990). In that region fog is also a major source of humidity (Dawson 1998, Burgess and Dawson 2004). In tropical rainforests, where annual precipitation exceeds 2400–2500 mm, tree heights of 60-70 m are not unusual.

1.2. Water use of trees Transpiration at tree level

Transpiration is an inevitable consequence of photosynthesis, the most important process of plants. When stomata in leaves are open for carbon dioxide uptake they allow water molecules to escape. The primary reason for transpiration is the diffusion of water vapor molecules from the intercellular air spaces of leaf (RH » 100%) through the open stomata to ambient air (Figure 4).

Driving force of the diffusion is the concentration difference, which can be up to 1.1 mol m^-3 in the case of water and about 0.0045 mol m^-3 of carbon dioxide. Taking into ac- count also the ratio of diffusitivies of these gases in air (Dw/Dc » 1.6, Lawlor 1993, Dw is

[H2O]»10 g m^-3 Figure 4. A schematic illustration of the structure of a leaf and the carbon dioxide (green arrow) and water (blue arrow) fluxes between leaf air space and the ambient air.

[H2O]»17 g m^-3 (at 20°C) - 32 g/m^-3 (at 30°C) mesophyll

cells

[CO2]»0.5 g m^-3

[CO2]»0.7 g m^-3 lower

epidermis

boundary layer

xylem upper

epidermis

parenchyma cells

vascular bundle

stoma

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the diffusion coefficient of water vapor in air and Dc is that of carbon dioxide) we obtain that for each carbon dioxide molecule harvested from air up to 400 molecules of water may be lost. This quite substantial transpiration per unit leaf area together with a total leaf area up to hundreds of square meters produces a significant flux of water at the tree level. The diurnal water use of a single tree can exceed 1000 kg (For a review, see Wullschleger et al.

1998). The vast majority of the water flowing in a tree is transpired in the air; less than 1 % is used in photosynthesis. In photosynthesis trees fix atmospheric carbon for energy resources and material growth. Allocation of photosynthates determines the sizes of the structural compartments (needles, branches, stem, coarse roots and fine roots) of trees (Nikinmaa 1992).

Transpiration induces sap flow, where liquid water is pulled from the soil through the stem and branches, up to the leaves where it evaporates in the air. The roots uptake water and dissolved nutrients from the soil and the wooden structure serves as a pathway for water flow from the roots to the shoots. Soil water availability, hydraulic conductance of the

Allocation

Water flow

Transpiration

Water transport capacity ofsapwood Embolism of conduits

Water uptake capacity of fine roots

Fine root conductivity Sapwood structure CO2 uptake for

photosynthesis

Sapwood

Fine roots Leafs

Leaf water Stomatal control

Fixed carbon

Soil water

Figure 5. A conceptual model of water related functions of a tree. State variables (amounts of leafs, sapwood, fine roots) are presented with thick-lined text boxes. Material flows are presented as blocked arrows with gray text boxes. Dashed curved lines with solid arrowheads represent influences.

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transport pathway and evaporative demand of the ambient air influences the water status of the leaves, which, in turn, can impose a limitation to gas exchange by controlling the stomata (Hari et al. 1999). Stomatal control has an effect on the carbon gain and thus on the growth of the tree. Transpiration, photosynthesis, carbon allocation and growth, tree structure, and water flow are all linked to each other (Figure 5). The hydraulic pathway has to be constructed in such a way that it can feed the necessary amount of water to the leaves.

The pipe-model theory of plant form (Shinozaki et al. 1964) states that tree stem consists of a bundle of unit pipes that connect functional roots to foliage units. Non-functional pipes form heartwood. The theory proposes a linear relationship between sapwood cross- sectional area and foliage mass or area. This linear relationship has been widely used as a principle to distribute resources between stem and foliage in tree growth models (see review by Sievänen et al. 2000).

Transpiration at stand level

Transpiration consumes substantial amount of energy because the heat of vaporization of water is quite high (2260 kJ kg^-1). This makes the transpiration of trees an important com- ponent in the energy balance of a stand and thus in the micrometeorology of the lowest part of the atmosphere that is called planetary boundary layer (PBL). PBL is, by definition, the part of the atmosphere, in which the Earth’s surface affects on the movement and the properties of air. The energy balance of a stand is:

LE H Q S G

R_n − − − = + (1)

where Rn is the net radiation, G the heat flux into the soil substrate, S the rate of change of heat storage (air and biomass) between the soil surface and the top of canopy, Q the sum of all additional energy sources and sinks, H is the sensible heat which is the heat absorbed or transmitted by a substance during a change of temperature which is not accompanied by a change of state, and LE is the latent heat which contains the energy stored in water vapor in transpiration and evaporation. The contribution of canopy transpiration to the total evapotranspiration of a stand (consisting of canopy transpiration, ground vegetation transpiration and evaporation from soil) depends on the stand structure (tree species, stand density, leaf area) and on the climatic factors, and it typically varies between 30% (Jiménez et al. 1999) and 65% (Grelle et al. 1997) of the total evapotranspiration. In addition, evapotranspiration of a forest stand exceeds evapotranpiration of a clearing in the same climatic conditions.

This is due to the transpiration from the forest canopy (Rannik et al. 2002). The increase of the ratio of sensible heat to latent heat (Bowen ratio) disturbs the rate of growth of the PBL (Culf 1992). Typically in boreal forests the Bowen ratio is bigger than in temperate forests leading to deeper PBL with lower water vapor concentration. This higher water vapor pressure deficit (VPD) reinforces a negative feedback on stomatal opening and transpiration and a positive feedback on sensible heat exchange (Baldocci and Vogel 1996).

The climate models used in the climate change research are very sensitive to the parti- tioning of energy between sensible heat and latent heat (Lundblad et al. 2001). In addition, water vapor is the most important greenhouse gas because it absorbs long wave radiation emitted by Earth. For these reasons, a better understanding of the interactions between the vegetation and the hydrological cycle is essential.

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1.3. The pathway for sap flow: the xylem

The stem of a coniferous tree consists of the following parts (Figure 6):

• Pith, a result of primary growth of apical meristem.

• Xylem, result of functioning of the vascular cambium. Consists mainly of axial tracheas. The newest part of xylem is called sapwood, which serves as pathway for water and nutrient transport. Sapwood contains living ray cells. The oldest part of xylem where rays cells are dead is called heartwood.

• Cambium, an extremely narrow layer consisting of few cells only, the lateral meristem, which increases the girth of the stem. Produces xylem and phloem. Located between xylem and phloem.

• Phloem, a narrow layer consisting of living cells which transport assimilates from leaves to growing tissues.

• Bark, prevents water evaporation from phloem. Protects the stem.

Every year cambium produces a new layer of xylem, an annual ring. The annual ring consists of axial tracheids and resin ducts and radial rays. Tracheids, which are 1… 4 mm in length and tens of micrometers in diameter, die in the same year and the empty cell lumens serve as a pathway for sap flow. In the beginning of the growing period cambium produces tracheas with thin cell walls and large cell lumen (early wood, lighter area of an annual ring; later on it produces tracheas with thicker walls (late wood, darker area of an annual ring) (Figures 6 and 7). The radial rays serve as path for horizontal transport of water and assimilates. The oldest annual rings in which the cells in radial rays are dead do not

Pith Heartwood

Sapwood

Bark

Figure 6. The structure of xylem in Scots pine. Photo: Albert Porcar, De- partment of Forest Ecology, University of Helsinki

Phloem Xylem

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transport water. This innermost part of the xylem is called heartwood. The newest annual rings constitute sapwood (Figure 6).

Tracheid

Figure 8. A microscope photo of a tangential cut of a Scots pine stem.

The groups of cells perpendicular to the longitudinal tracheids are rays with living cells. Photo: Tuuli Timonen, Botanical Museum, University of Helsinki.

Rays Annual ring

Early wood

Late wood Resin duct

Figure 7. A microscope photo of a stem cross-section of a Scots pine.

Photo: Tuuli Timonen, Botanical Museum, University of Helsinki.

Ray

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The ends of the tracheids overlap in the vertical direction (Figure 8) and water flows from one tracheid to another via small pores called interconduit pits. The structure of pits varies between coniferous species. In Scots pine a pit contain a torus and margo, which is a net around the torus (Figure 9). Interconduit pits are the check valves that permit water flow between conduits but inhibit the leaking of air into the transpiration stream (Dixon 1914 cited by Sperry 2003). When there is a pressure difference between cells, the aspiration of torus against the pit aperture seals the pit and prevents the growth of gas bubble to another cell lumen.

Cell wall

Figure 9. A microscope photo showing the interconduit pits in the tracheid wall of Scots pine and a schematic illustration of a pit structure with torus (black centre) and margo (surrounding net-like structure). Photo: Tuuli Timonen, Bo- tanical Museum, University of Helsinki.

Torus

Margo

Middlelamella

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1.4. From transpiration to water tension and sap flow The physical basis for water ascent

in plants derives from the structure of a water molecule, where an oxygen atom and two hydrogen atoms form an asymmetrical composition (Figure 10). This asymmetrical composition with opposite electrical charges of oxygen and hydrogen ions lead to dipolar structure, which enables water molecules to attach to each other (cohesion) and form clusters of molecules. Adhesion is the process where water molecules are at- tracted to polarizable or charged surfaces of other substances (wettable).

Water flow through sapwood of a tree is initiated in intercellular air spaces of leaves from where water molecules diffuse to ambient air (Figure 4). This causes the net evaporation of water molecules from water surfaces on parenchyma cells and increases the concavity of the surface of the water film between cells (Figure 11) or into pores in cell wall. Surface tension (relatively high for water, 0.073 N m^-1), which results from asymmetric forces between water molecules at the surface, tends to

minimize the area of the surface and pulls the water molecules towards the surface. This results in a pressure drop in the water below the concave surface. The pressure drop is a function of the radius of the curvature and the surface tension of water (Laplace equation (Pickard 1981)):

p r

p ²γ

0−

= (2)

where p is the pressure under curved surface of water, p0 is the ambient pressure, g is surface tension of water, r is the radius of the concave surface. The pull i.e. tension between water molecules propagates along the continuous water chain down to soil and causes them to rise through the tracheary elements of the stem. The hydrogen bonding between adjacent water molecules keep the molecules together as a continuous chain. This is the cohesion –

O^2-

H⁺ H⁺

104°

Figure 10. A schematic illustration of the composition of a water molecule.

p0

r

p

Figure 11. Schematic illustration of curved surface of water film on parenchyma cells inside a leaf.

water film

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tension theory of sap flow in trees, which was presented already in 1894 by H.H Dixon and J. Joly (Pickard 1981).

The cohesion theory implies that a tension gradient of 0.02…0.03 MPa m^-1 should exist inside xylem when transpiration is present (0.01 MPa m^-1 due to gravity and 0.01…0.02 MPa m^-1 because of frictional pressure losses in flow). This leads to a tension of two megapascals at the top of tall trees. This predicted tension is the reason why the cohesion – tension theory remained controversial for decades: nobody had demonstrated the existence of negative pressure of water in plants and how is it possible that water columns can remain continuous under high tension for long periods of time without breaking into separate molecules. This intuitively unavoidable phase change should lead to embolism, where the conduit is filled with gas and water flow is blocked.

The existence of tension was shown at the first time in 1965 when Scholander and his co-workers introduced the pressure bomb (Scholander et al. 1965). In a pressure bomb positive pressure is used to push water out from the cut end of a leaf or a twig. The pressure at which water starts to flow out from the leaf corresponds to the negative pressure or tension inside the leaf. Tensions down to –17 MPa have been measured by the pressure bomb (Kappen et al. 1972 cited by Zimmermann et al. 2004). Later on Holbrook et al. (1995) and Pockman et al. (1995) using a centrifugal technique showed maximum sustained water tensions ranging from -1.2 to -3.5 MPa in stem segments of various species.

1.5. Implications of water tension

1.5.1. Stem sapwood diameter changes

While connected to each other with cohesive forces, the molecules in a stretched water column are attached also to the walls of tracheary elements with adhesive forces. With this mechanism the tension between water molecules is transferred also to the elastic cell walls and they bend inwards and the diameter of cell shrinks. At the scale of a stem (bundle of individual cells) this is added up to sapwood diameter changes of detectable order of mag- nitude (Irvine and Grace 1997).

Diurnal reversible changes of tree stem diameter have been detected since the early 20^th century (MacDougal 1924). In the early days the variation of stem diameter was considered as variation in the water content in bark and phloem. Later on also variation in xylem diameter has been observed, and at the present time it is considered as a direct and immediate indicator of transpiration-induced water tension fluctuation in xylem (Irvine and Grace 1997).

1.5.2. Embolism

A phase diagram (Figure 12) shows the stable preferred physical states of water at different temperatures and pressures. At typical room temperature and pressure water is a liquid, but it becomes solid (i.e. ice) if its temperature is lowered below 0 °C, and gaseous (i.e. water vapor) if its temperature is raised above 100 °C, at the same pressure. Each line gives the conditions when two phases coexist but a change in temperature or pressure may cause the phases to abruptly change from one to the other.

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The saturated vapor pressure of water is the partial pressure of water vapor that is in equilibrium with liquid water at a particular temperature.

When saturated vapor pressure equals ambient pressure, bubbles of water vapor are formed in liquid water. If this is achieved by increasing temperature at constant pressure, the process is called boiling; if de- creasing pressure at constant temperature it is called cavitation (Bren- nen 1995). If phase-change has not occurred when pressure-temperature combination has gone outside the stability region in the phase diagram, water is in metastable state (super- heated (solid/liquid) or supercooled (liquid/vapor). Thus, liquid water,

which is pulled upwards through the xylem according to the cohesion-tension theory, and which pressure is lower than 0.023 MPa (at 20 °C) is in metastable state.

Cavitation is a special case of nucleation, which is the process of spontaneous formation of a new phase within the volume of the pre-existing phase by random fluctuation. Phase change of a substance is explained by the classical nucleation theory (CNT) (Brennen, 1995). In cavitation, gas bubbles are formed in the liquid in regions of low pressure with constant temperature. The tiny nuclei of the new phase tend to grow rapidly if the size of the nuclei exceeds certain critical size. If the size of the nuclei is below the critical size surface tension will dissolve them. The probability of forming a void of critical size is dependent on the tension and the temperature of the water. The CNT distinguishes two types of nucleation: homogeneous and heterogeneous. In homogeneous nucleation the thermal mo- tions of molecules within the pure liquid form microscopic voids, which can grow, but the tension for breaking the pure liquid water is hundreds of megapascals and far beyond the tensions in xylem water. In heterogeneous nucleation, interactions between water molecules and impurities or surfaces of other substances enable the formation of nuclei with critical size even at moderate tensions. Cavitation is a stochastic process and the probability of a cavitation event increases with time, hydrophobicity of the surface, and hydrophobic surface area. Wettability of a surface is expressed as the contact angle α, which a water droplet forms with a surface. If α < 90° the surface is wettable i.e. hydrophilic, the adhesive forces are dominating; if α > 90° the surface is non-wettable i.e. hydrophobic, the cohesive forces between water molecules are dominating (Pickard 1981) (Figure 13.). A single successful cavitation event with bubble growth is enough to embolize the whole conduit.

According to the nucleation theory, nucleation sites, a nucleation event, and bubble growth are required for phase-change from liquid to gas. Hence, the metastable state of liquid water is possible when conditions for nucleus formation and bubble growth are in- adequate, although pressure and temperature would predict a phase change by cavitation according to the phase diagram of water.

In ecophysiological literature, the term cavitation is widely used as a synonym for em- bolism although it is only one mechanism leading to embolism, which means the occlusion

Figure 12. A phase diagram of water.

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of a conduit by a gas bubble. In cavitation, a phase change creates the initial nucleus of the gas bubble. In two other mechanisms leading to embolism, an existing gas bubble works as an initial nucleus for bubble growth.

In air-seeding (Zimmermann 1983), tension draws an air bubble through a pore or pit in the cell wall into the cell lumen, where water evaporates into the bubble which then en- larges and fills the whole conduit. The threshold value of pressure which leads to air- seeding is a function of pore size and surface tension of water (Laplace’s equation, Eq. 2.).

The third mechanism leading to embolism is the release of existing bubbles from cracks or crevices of the cell walls (Pickard 1981). Vapor bubbles most efficiently stabilize by surface tension forces in steep non-wettable cracks or crevices of the cell wall (Figure 13).

The bubble is released into the liquid when the pressure difference Dp between the bubble and the water equals to (Pickard 1981):

(

α β

)

γ −

=

∆ 2 cos rc

p (3)

where _a is the contact angle of the three phase interface, _b is half of the apex angle of the conical crevice, and rc is the radius of the crevice.

Air-seeding and release of existing bubble are deterministic processes: when tension has reached some threshold limit the initial nucleus starts to grow and the conduit is filled with gas.

The nucleation processes described by the CNT set a limit to the strength of water to

cope with tension. Theoretical tensile strength of pure water is -140 ... -230 MPa (Pickard 1981), but as mentioned above the existence of heterogeneous nucleation lowers substan- tially the maximum sustained tension. In addition, the two other mechanisms leading to embolism take place under considerably lower tensions according to the equations above.

1.5.3. Prevention of embolism

Embolism causes decrease in hydraulic conductivity by blocking the conduit with a gas bubble and therefore restricts sap flow, which, in turn, increases the water tension. It has been suggested, that this may eventually lead to ‘runaway cavitation’ (Tyree and Sperry 1988). On longer run, it is crucial that the excessive amount of embolism can be avoided in order to maintain the water transport capacity. As trees grow taller it is even more important because longer transport pathway increases the water tension and makes the embolism more apparent.

During evolution xylem of higher plants has undergone structural adaptation to avoid excess emboli formation (Sperry 2003). The cell lumens are small (cavitation probability increases with cell surface area). Xylem walls are air-free (Zimmermann 1984), or air bubbles are unlikely in newly produced conduits (Sperry 2003), and water uptake by roots ex- cludes air bubbles (Zimmermann 1983). Lignified and thickened cell walls are able to avoid the wall collapse under tension (Hacke et al. 2001). Cell wall pores are small (1.2…3.3 nm) enough to prevent air-seeding under tension lower than –15 MPa (Sperry 2003). Most of Figure 13. A schematic illustration of a gas bubble stabilized in a conical crevice in the cell wall.

2b

liquid a

gas bubble

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the conduit walls are highly wettable because wettability of cellulose and hemicellulose, the main constituents of the conduit wall, is high (Pickard 1981). The walls contain also hydrophobic lignin, which may destabilize the xylem water under tension, but its concentration in cells of conifer xylem is highest in the middle lamella and drops to lower values closer to the lumen (Donaldson 2001). The interconduit pits are able to block the expansion of a gas bubble from one conduit to another by aspirating the torus against the pit aperture (Sperry 2003).

There are several structural adaptations to avoid hazardous water tensions. The decrease of the leaf area to sapwood area ratio with increasing stem height (McDougall et al. 2002, Vanninen et al. 1996) increases the leaf specific hydraulic conductance of the tree and de- creases the maximum water tensions at tree top. The production of xylem vessels with increased permeability (Pothier et al. 1989), an increase in the fine root foliage ratio (Sperry et al. 1998; Magnani et al. 2000), and increased water storage in the stem (Phillips et al.

2003) also lead to lower water tensions.

The functional way to avoid excess embolism is to regulate the transpiration induced water tension by partial closure of the stomata. This leads to reduced CO2 uptake and lower photosynthesis. This is typical in sunny afternoons when evaporative demand exceeds water uptake and transport capacity of the tree leading to reduced gas exchange (Figure 14).

This can be seen in the asymmetric CO2 exhange although photosynthetically active radiation, PAR (400-700 nm), which is the driving force of photosynthesis, is symmetric around noon.

1.5.4. Indication of embolism

Although trees have evolved to avoid excessive embolism, the emboli formation has been found to be a regular occurrence (Perks et al. 2004). According to the current knowledge air-seeding from an already embolized conduit is the most probable mechanism of embolism. It has been proposed that with high enough water pressure difference between a gas- filled conduit and a conduit containing stretched water, the torus edge of the interconduit pit structure is pulled through the pit aperture and air-seeding occurs (Sperry and Tyree 1990, Sperry 2003).

During embolization plants produce detectable acoustic (Milburn and Johnsson 1966) and ultra-acoustic (Tyree and Dixon 1983) emissions, which are thought to result from the shock wave following a cavitation or air-seeding event (Jackson and Grace 1996). These emissions can be recorded with ultra-sonic acoustic sensors (Hölttä et al. 2005). Figure 15

-0.04 0 0.04 0.08 0.12 0.16 0.2

0:00 6:00 12:00 18:00 0:00

Time (hh:mm) CO2 exchange (mg m -2 s-1 )

0 500 1000 1500

PAR (µmol m-2 s-1 )

Figure 14. Measured CO2 exchange (solid line) of a Scots pine shoot enclosed in a cuvette and PAR (dashed line) in Hyytiälä 5.5.2004.

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presents the time course of ultra-acoustic emissions recorded from a Scots pine stem in Hyytiälä during two days in August 2000. Figure 15 illustrates also microvariation of xylem diameter which reflects the tension fluctuation due to changes in transpiration as ex-

plained above in section 1.5.1.

1.5.6. Repair of embolized conduits

Embolism seems to be a usual occurrence in tree stems during periods of increasing transpiration. Because gas-filled cells cannot transmit tensions, embolized conduits are lost from the water transport system. To maintain hydraulic capacity, plants must replace embolized cells, maintain a highly redundant transport system, or repair embolized conduits (Holbrook and Zwieniecki, 1999). Pickard (1989) has suggested that embolized conduits might be restored to their functional state but this restoration has usually been connected to situations where the whole plant has been pressurized by root pressure (Tyree et al., 1986; Cochard et al., 1994; Fisher et al., 1997). In recent studies, it has been suggested that embolized conduits may be repaired also, when the water in neighboring conduits is under tension (McCully et al., 1998; Zwieniecki and Holbrook, 1998; Tyree et al., 1999; Melcher et al., 2001). Holbrook and Zwieniecki (1999) suggested that living cells in xylem parenchyma provide the driving force for refilling the embolized conduits when they are hydraulically isolated. This implies that the pressure of water in the refilling vessel is equal to the bubble gas pressure.

Later on, Holbrook et al. (2001) presented direct observations of xylem embolism and embolism repair in an intact grapevine stem by using a magnetic resonance imaging (MRI) technique. They detected about 10 individual emboli formations during a 24-hour period of active transpiration. Re-watering the plant relaxed the water tension rapidly but embolism recovery did not start until turning off the lights stopped transpiration and sap flow. Al Figure 15. Stem xylem diameter (darker line) and the observed ultra- acoustic emissions (lighter line) at the height of 2.5 m of a Scots pine stem in Hyytiälä for 13.8.- 14.8.2000. From Hölttä et al. (2005).

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though grapes are well known for their capacity to produce root pressure (Sperry et al.

1987) Holbrook et al. did not find any indication of this as root exudation.

1.6. Measurement of water flow

Sap flow in a tree is purely a physical phenomenon and its physics is fairly well understood (Pickard 1981). However, its quantification methods in intact trees are not problem-free.

Sap flow in tree stems has been measured since the pioneering work of Huber in the early decades of the 20^th century (see review by Čermák et al. 2004). The main methods used in the field apply thermodynamics: heat pulse velocity HPV (Huber 1932, cited by Čermák et al. 2004), trunk segment heat balance THB (Čermák and Deml 1972), stem heat balance SHB (Sakuratani 1981), heat dissipation HD (Granier 1985), and the most recent heat field deformation HFD (Nadezhdina et al. 1998, Nadezhdina and Čermák 1998).

The THB method is directly quantitative and needs no calibration; flow is calculated from the applied energy, the temperature change, and the specific heat of water. The THB method has been validated by volumetric techniques on several tree species. Flow rate, scaled up to stand transpiration, has shown good agreement with chamber measurements at branch level and evapotranspiration measured by the eddy-covariance method during dry conditions for a stand (Lundblad et al. 2001). The THB method is very robust and provides reliable data during long-term measurements in trees with diameters over 15 cm for a broad range of tree species, sizes and environmental conditions. It has been applied as a standard when testing other methods (Offenthaler and Hietz 1998; Nadezhdina and Čermák 1998;

Lundblad et al. 2001).

The SHB method is based on the heat balance of a stem section, which is heated by a heating belt wrapped around it. The temperature increase over the heated section, the radial and vertical heat fluxes are measured, which together with the input power, are used when calculating the sap flow. It has been proposed (e.g. Shackel et al. 1992) that the SHB method is not appropriate for thick stems.

The HD is a relative method based on heat dissipation around a heated probe, and was developed from empirical laboratory calibrations (Granier 1985, 1986). Originally it was calibrated for five tree species and sawdust and it was assumed to be valid for all tree species. Later, studies have found that calibration should be done separately for each species (Smith 1996).

The HFD method is based on measurements of the deformation of the heat field around a needle-like linear heater inserted in a radial direction into the stem. The frontal view of the heat field under zero flow looks like a symmetrical ellipse due to different heat conductivity of the stem in axial and tangential directions and obtains a form of a gradually pro- longing deformed ellipsoid under increasing flows. Sap flow using the HFD method is calculated from the ratio of temperature gradients around the linear heater in the axial and tangential directions. Experimentally, it was found that this ratio is proportional to the sap flow rate (Nadezhdina 1998, Nadezhdina et al. 1998).

All the above techniques have problems: either they have to be calibrated for different tree species (HD), they lack a sound physical basis and are based exclusively on empirical observations only (HD, HFD), or they have restrictions with stem size (THB, SHB).

1.7. Dynamic modeling approach

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The theories of classical physics can explain phenomena in the natural environment with considerable accuracy. One could say that these theories have received a high measure of corroboration in a Popperian sense (Popper 1959). These phenomena include for example the movement of particles in a gravitational field, mass, energy and momentum transfer, and behavior of flowing fluids. They all obey basic physical principles like conservation of mass, energy and momentum. Sap flow and related phenomena are part of the physical environment and they can be studied by using methods of physics. The basic processes behind the phenomena can be modeled using mathematical equations with physical interactions and quantities, and the models can be implemented as computer programs.

Giere (1988) suggests that theoretical models are means by which scientists represent the world – both for themselves and for others. A model is an abstract entity that represents a system found in the real world. Models, as the statements defining them, come in varying degrees of abstraction. The relationship between model and real system is similarity. Hy- pothesis claim similarity between models and real systems, and as linguistic entities they can be true or false. A scientific theory includes both statements defining the models and hypotheses claiming a good fit between the models and some important types of real sys-

tems (Figure 16).

Term dynamic refers to phenomena that produce time-changing patterns (Luenberger 1979). A dynamic system is a system whose state varies within time. Systems with flow of some substance are typical dynamic systems. A dynamic model is a model describing the time evolution of a state of the dynamic system.

The dynamic modeling approach gives us an opportunity to study functions of compli- cated systems. With models based on real physical interactions, it is also possible to analyze phenomena, which are difficult to measure directly. According to Bossel (1994), a key question in modeling is the structure of the model. State variables and processes should have actual counterparts in the physical world. The values of the model parameters should be obtained from direct and independent measurements.

The cohesion theory described in section 1.4. explains the water movement at the molecular level stating that individual water molecules are connected to each other by cohesive forces and these water chains are pulled upwards to the transpiring leaves. This molecular level is, however, far too detailed for quantitative modeling of sap flow and related phenomena and a more approximate approach is needed.

similarity definition

Set of statements

Real system

Figure 16. Models and real systems according to Giere (1988)

Model

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1.8. Other models in the literature

Water flow in a single tree has been modeled in several studies (Edwards et al. 1986, Tyree 1988, Früh 1997). All of these models are based on the capacitance analogy: the relation between water pressure p and amount of water θ is capacitance dθ⁄dp (change in water content divided by change in pressure). The problem with this approach is that the model results are not directly comparable with field measurements.

The performance of different sap flow measuring systems has been analyzed in several studies. Especially the SHB method has received considerable interest (Baker and Nieber 1989, Groot and King 1992, Grime and Sinclair 1999).

In the literature there are not many studies modeling embolism repair. Yang and Tyree (1992) considered the dissolution and movement of gas, and since their interest was the entire stem, they applied a 2-dimensional cylindrical symmetric model. However, they assumed that the pressure in the water adjacent to the bubble is equal to the stem xylem water pressure, i.e. hydraulic compartmentalization was not assumed.

2. THE AIM OF THIS STUDY

The principal aim of the present study is to extend our understanding of the dynamics of tree water relations. Five sub-studies were carried out during this process, in each a dynamic model based on known physical interactions was developed to analyze the water relations in tree stem. Sap flow and water pressure behavior was examined in StudiesI, II and V. A particular objective of these works was to formulate sap flow model in a way that facilitates the direct comparison of simulated results with field measurements. The effect of embolism was studied in Study II. The aim of Study III was to examine under which conditions embolism repair can take place assuming that the embolized vessel is hydraulically isolated, like Holbrook and Zwienicki (1999) have suggested. In Study IV we examined the performance of a sap flow measuring system based on the SHB method.

The fundamental assumption behind these models is that the studied objects and phenomena are parts of the physical environment and physical interactions hold true. From physics we obtain the elementary processes in the models: convective flow of water is driven by pressure difference, diffusion of a substance is driven by concentration difference, and heat energy is transported with convective flow of sap and conduction, which is driven by temperature differences. Elastic material (wood) is shrinking under pressure. The models themselves are sets of mathematical equations describing the processes changing the state of the systems in time.

In sap flow models (Studies I, II and V) and in the model analyzing the Dynamax™ sap flow measuring system (Study IV), sapwood of tree stem is treated as a homogenous (in radial direction) porous material, with specific axial permeability for water. In the model for embolism refilling (Study III), the object of study is treated as a combination of individual cells. In addition, in the embolism sub-model of Study II the cell properties, which affect on the embolism formation, are considered as distributions. Water is treated as a fluid

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having properties as: mass, density, pressure, viscosity and temperature. In Study III, where water is treated as a solution, also concentration of sugar, is considered.

The main hypotheses tested in this work were:

• Transpiration creates curved water surfaces inside leaves. The concavity of the surface causes a water pressure drop, which rapidly propagates through the stem sapwood as predicted by the cohesion-tension theory. Pressure gradient causes the sap flow.

• Xylem pathway structure causes time lags in transport.

• The physical processes leading to embolism and embolism recovery are feasible within the normal tension variation and woody structure.

• Sap flow mechanism imposes structural limitations on tree stem.

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3. MATERIALS AND METHODS

3.1. Description of the models in this study 3.1.1. About the modeling methodology The starting point for modeling flow systems is to describe the system with a set of mathematical (differential or integro- differential) equations and boundary conditions. The equations are usually so compli- cated that they cannot be solved analyti- cally and a numerical solution has to be applied. The next step is to select the discretization method i.e. a method for ap- proximating the differential equations by a system of algebraic equations. There are two main approaches to select discretization methods: finite difference method (FD) and finite volume method (FV) (Figure 17).

The FD method is the oldest method for numerical solution of partial differential equations. The starting point is the conservation equation in differential form. The solution domain is covered by a grid and at each grid point the derivatives are replaced by approximations. This results in a group of algebraic equations which can be solved (Ferziger and Peric 1996). An example of the FD methods is the Crank-Nicolson method for solving parabolic partial differential equations.

The FV method starts from the integral form of the conservation equations and they are applied into a finite number of contigu- ous control volumes of the solution domain.

An example of these methods is the QUICK method (Ferziger and Peric 1996).

3.1.2. Sap flow models

Models describing sap flow (Studies I, II and V) assume that the sapwood of the tree stem is a homogeneous water-conducting elastic material and the diameter of sapwood under- goes diurnal shrinking and swelling caused by transpiration-induced fluctuating pressure i.e. tension (p<0) of stem sap. The change in the stem volume is assumed to match the change in water volume. Sap flow is driven by pressure gradient (Darcy’s law):

Physical background

Appl. for finite volume element Partial differential

equations

Approximation in grid points

Algebratic equations

Numerical solution

Approximate solution

FD FV

Figure 17. Discretization methods Algebratic equations

Numerical solution

Conservation equations in integral form

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η ^dh^Aρ dp

Q=−k (4)

where Q is sap flow, k is sapwood permeability, η is the dynamic viscosity of water, p is water pressure, h is length of stem segments, A is basal area of sapwood and, ρ is water density. Radius of the stem is changing with change in pressure (Hooke’s law):

( )

r hw

E r r dp

dr = − (5)

where r is stem radius, r_hw radius of the heartwood and E_r is the radial elastic modulus (elasticity) of sapwood.

The mass balance of stem segment:

E Q dt Q

dm

i out i in i

w, = , − , − (6)

where mw,i is the water mass in segment i, t is time, Q_in is the inward mass flow rate and Q_out is outward mass flow rate and E is transpiration from the segment. In StudiesI and II the above equations where applied to a chain of elements of finite size for a short time step dt. The solution method can be regarded as a FV method.

In Study V the same physical background was used in the formulation of a single diffusion-advection-reaction – type of partial differential equation, which describes water pressure propagation inside a tapering (∂r/∂h) tree stem with a transpiration density S:

2 2

2 1 h

p r r k E t

p _r _hw

∂ ú∂ û ê ù

ë

é ÷

ø ç ö è æ +

∂ =

∂

η ⁽⁷⁾

(

^r^E ^k^r

)

^h^r ^k ^h^p ^r

(

^r^r ^E^r ^k

)

^r^h ^g^k ^h^p

h k r r

E _hw

hw r hw hw p

r hw

r

∂

∂ úú û ù êê

ë

é +

∂

− −

∂ + ∂

∂

∂ + −

∂

÷∂ ø ç ö è æ +

+ ρ η

η η

η η ¹

2

(

^r^E ^r^gk

)

^h^r ^r^E

(

^r ^gkr^r

)

^r^h ^r

(

^r^E ^r

)

^S

h k r r g E

hw r hw

hw hw r hw p

r hw

r

− −

∂

− −

∂

∂ + −

∂

÷∂ ø ç ö è æ +

+ η π

ρ η

ρ 1 2 2

where ρ is water density and g is the gravity. This equation denotes that pressure is propa- gating by ‘diffusion’ (diffusion coefficient corresponds to the multiplier of the second spatial derivative of pressure) and ‘advection’ (advection velocity is the multiplier of the first spatial derivative of pressure) along the stem, and is depleted according to the sink term (last line in equation 7; usually negative). The diffusion term equals Ek/2η in a stem without heartwood and approaches Ek/η with increasing heartwood percentage. The advection Figure 18. A schematic illustration of a stem segment showing transpiration, sap flow, and diameter shrinkage.

E

Qin

Qout

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term introduces the effects of change in permeability (∂k/∂h), stem taper (∂r/∂h) and (∂rhw/∂h), and the gravity (ρgk/η). The sink term originates from transpiration and changes in water conduction capability of the stem (∂k/∂h, ∂r/∂h, and ∂rhw/∂h). First, the FD method (Crank-Nicolson) was applied to solve the Equation 7, but it pointed out to be unstable because the advection term is too big compared to the diffusion term. Later on, the FV method (QUICK scheme) was used to obtain the approximate solution.

3.1.3. Sap flow and embolism model

In Study II the sap flow model in Study I was extended by adding an embolism sub-model and the effect of embolism on sap flow, sapwood permeability, and sapwood diameter. The embolism sub-model included three mechanisms for emboli formation: heterogeneous cavitation, air-seeding and the release of pre-existing bubbles from cell wall crevices. Het- erogenous cavitation is covered by the following equations (e.g. Blander and Katz, 1975):

( )

úú

û ù êê

ë é

−

= 0 − ³ ₂

3 exp 16

l

v p

p kT J F

J πγ (8)

where J is the nucleation rate giving the number of nucleation events per unit area of the foreign surface (conduit wall or impurities) and unit time. J0 is a kinetic pre-factor proportional to N^2/3, where N is the molecular density, k is the Boltzman factor, T is the temperature, (pv – pl) is the difference between saturation vapor pressure and liquid pressure, re- spectively, and F is a factor dependent on the contact angle:

4 cos cos 3

2+ α − ³α

=

F (9)

where α is the contact angle. The probability that a conduit still contains liquid after time t is e^-JtAh, where Ah is the hydrophobic surface area of the cell lumen. Hence, the probability for cavitation is (e.g. Koop et al., 1996)

e JtAh

P=1− ⁻ (10)

Parameters that affect embolism vulnerability: the wettability of conduit surfaces (contact angle), sizes of the single largest pores in the intervessel pits, and sizes of the largest crevices in the conduit walls were treated as distributions.

Air-seeding is covered by Equation 2 and the release of pre-existing bubbles is covered by Equation 3.

3.1.4. Embolism recovery model

Holbrook and Zwienicki (1999) suggested that embolism recovery (i.e. refilling of embolized conduit) is possible in conditions where water in the adjacent xylem conduits is under tension. According to their hypothesis living cells adjacent to the embolized vessel would create a driving force that draws water into the vessel lumen pressurizing the gas, which tends to dissolve into the xylem fluid and further transported away by diffusion.

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Starting from their hypothesis a model describing the above processes as physical phenomena was developed (Study III).

The model was based on the following assumptions: In a hydraulically isolated refilling vessel, water pressure is equal to bubble gas pressure. Source of water for refilling is from water conducting vessels via living cells under turgor. Continuum of living cells act as source of solutes. Turgor driven water inflow pressurizes gas that dissolves into water and diffuses away from the embolized cell. In addition we assumed that reflection coefficient between embolized and living cell is below unity (solute transport between those cells is allowed), while the reflection coefficient between living cells and waters conduction cells equals unity (the solute transport is prohibited). The reflection coefficient represent the cell membrane aq- uaporins which are structures consisting of hydrophobic proteins (Tyerman et al. 2002), and there is strong evi- dence that they are central components in plant water relations. Figure 19 presents the system and the processes covered by the model. For detailed model equations see Study III.

With this model, refilling process of the embolized conduit was examined in the range for the values of physico-physiological quantities (water tension in xylem, osmotic potential of the living cells, and diffusion distances).

3.1.5. Model for analyzing a sap flow measuring system

In Study IV a model was developed to analyze the performance of Dynamax Flow32 sap flow measuring system, which is based on the stem segment heat balance (SHB) method.

The model simulates the heat propagation by conduction and convection (with a prescribed sap flow) in a stem segment, which is heated by a heating belt wrapped around the stem.

Additionally, the model calculates sap flow from simulated heat transfer according to the SHB method.

Figure. 19. Schematic representation of the system of the refilling vessel (1), water- conducting vessels (2), living cells (3), ray cells (4) and phloem (5). The refilling vessel is hydraulically isolated from the xylem vessel. The refilling vessel contains liquid water and gas. The air dissolves in the water and diffuses out of the vessel. From Study III.

A physical analysis of sap flow dynamics in trees

Martti Perämäki

Department of Forest Ecology, University of Helsinki, Finland

Academic dissertation

ABSTRACT

ACKNOWLEDGEMENTS

LIST OF ORIGINAL ARTICLES

TABLE OF CONTENTS

LIST OF TERMS

LIST OF SYMBOLS AND CONSTANTS

1. INTRODUCTION

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2. THE AIM OF THIS STUDY

3. MATERIALS AND METHODS

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